The present invention relates to educational game apparatuses and methods for use in teaching the addition, subtraction, multiplication, and division of positive and, especially, negative numbers. The game apparatuses and methods of the present invention enable students to see and understand a theory for adding, subtracting, multiplying, and dividing positive and negative numbers.
A description of the prior art is set forth in U.S. Pat. Nos. 1,294,126, 3,094,792, 3,229,388, 3,410,002, 3,414,986, 3,452,454, 3,935,649, 4,177,681, 5,474,455, 6,089,871, and U.S. Pat. No. 6,413,099, which patents are incorporated herein in their entireties by reference.
As evidenced by the above-cited patents, educational game apparatuses and methods exist for teaching mathematical concepts. However, teaching the addition, subtraction, multiplication, and division of positive and, especially, negative numbers usually entails students learning by rote the rules of adding, subtracting, multiplying, and dividing positive and negative numbers without ever understanding the rhyme or reason behind what they are doing. Other students, unfortunately, never learn the rules and, for them, mathematics becomes a dreaded black hole.
Accordingly, a technique is needed for teaching the addition, subtraction, multiplication, and division of positive and, especially, negative numbers that clearly explains a cogent theory behind the rules.
The apparatuses and methods of the present invention for teaching the addition, subtraction, multiplication, and division of positive and negative numbers solve the above need. More specifically, the present invention is based on the Null Theory of Adding, Subtracting, Multiplying, and Dividing Positive and Negative Numbers (hereinafter referred to as the xe2x80x9cNull Theoryxe2x80x9d). According to the Null Theory, the natural state of a given environment is the null state. In the null state, the environment is in perfect balance and appears to be devoid of any matter. However, the environment is, in fact, composed of a plurality of null units, with each null unit being, in turn, composed of a positive unit and a negative unit. The environment can be disturbed by introducing (i.e., adding) into it one or more positive units or one or more negative units. The environment can also be disturbed by removing (i.e., subtracting) from it one or more positive units or one or more negative units. In the latter case, if there are not enough free positive units available to be removed from the environment, a sufficient number of null units are split to obtain the desired number of positive units to be removed from the environment. When a null unit is split and the positive unit thereof is removed from the environment, a negative unit is left behind in the environment. (Hence, the foregoing explanation of the Null Theory clarifies and visually demonstrates the reason behind the rule that the subtraction of a positive number +X is equal to the addition of a negative number (i.e., xe2x88x92(+X)=+(xe2x88x92X)).) Likewise, if there are not enough free negative units available to be removed from the environment, a sufficient number of null units are split to obtain the desired number of negative units to be removed from the environment. When a null unit is split and the negative unit thereof is removed from the environment, a positive unit is left behind in the environment. (Accordingly, the foregoing explanation of the Null Theory clarifies and visually demonstrates the reason behind the rule that the subtraction of a negative number xe2x88x92X is equal to the addition of a positive number (i.e., xe2x88x92(xe2x88x92X)=+(+X)).)
Another aspect of the Null Theory is that only excess positive units or excess negative units remain in the free state within the environment. For example, if there are 5 free positive units in the environment and if 3 negative units are introduced into the environment (as is the case in the mathematical expression 5+(xe2x88x923)), the 3 negative units will combine with 3 of the free positive units to form 3 null units, leaving only 2 free positive units in the environment.
With the Null Theory in mind, in one embodiment of the present invention, the game apparatus employed to teach the addition, subtraction, multiplication, and division of positive and negative numbers comprises (a) a plurality of positive units, (b) a plurality of negative units, and (c) a demarcated playing environment or zone. The positive units and the negative units are adapted to reversibly attach to or be associated with one another to form null units, with each null unit comprising at least one positive unit and at least one negative unit and the number of positive units and the number of negative units per null unit being equal. (As used in the specification and claims, the terms xe2x80x9cattached toxe2x80x9d and xe2x80x9cassociated withxe2x80x9d both mean that the objects in question either can be physically reversibly held together or can be positioned in a manner such that the objects appear to have an affinity for or relationship with one another.) Preferably, each null unit comprises just one positive unit and just one negative unit.
The demarcated playing zone is typically an integral part of a playing surface.
Generally, the game apparatus further comprising a first means for measuring the number of units selected from the group consisting positive units, negative units, and combinations thereof, with the first measuring means desirably being located on the playing surface and, preferably, within the demarcated playing zone. The purpose of the first measuring means is to measure the degree that the demarcated playing zone has been disturbed from the null state. The first measuring means, which can be a scale for weighting the positive and/or negative units, is ideally an axis marked in substantially equal units from 0 to M and in substantially equal units from 0 to N, where M is a positive whole integer, N is a negative whole integer, and substantially each of the positive units is adapted to reversibly attach to or be associated with a unit from 0 to M on the axis on the playing surface, and substantially each of the negative units is adapted to reversibly attach to or be associated with a unit from 0 to N on the axis on the playing surface. While M can be virtually any positive integer, M is typically a whole positive integer from 5 to 50, more typically from 10 to 25, and most typically from 10 to 20. Similarly, while N can be virtually any negative integer, N is commonly a whole negative number from xe2x88x925 to xe2x88x9250, more commonly from xe2x88x9210 to xe2x88x9225, and most commonly from xe2x88x9210 to xe2x88x9220. Usually, M equals the absolute value of N.
In another preferred embodiment of the invention, the apparatus further comprises a second means for measuring the number of units selected from the group consisting positive units, negative units, and combinations thereof, with the second measuring means desirably being located on the playing surface and, preferably, outside the demarcated playing zone. The purpose of the second measuring means is to act as a check point or zone to ensure that (a) the correct number of positive units and/or negative units are being transported into the demarcated playing zone and (b) the correct number of positive units and/or negative units have been removed from the demarcated playing zone. The second measuring means, which can also be a scale for weighting the positive units and/or the negative units, is ideally an axis marked in substantially equal units from 0 to P, where P is a positive whole integer and substantially each of the positive and negative units is adapted to reversibly attach to or be associated with a unit from 0 to P on the axis on the playing surface. While P can be virtually any positive integer, P is typically a whole positive integer from 10 to 100, more typically from 20 to 50, and most typically from 20 to 40. Usually, P equals M plus the absolute value of N.
It is also preferred that the game apparatus further comprise a plurality of means for reversibly holding a plurality of the null units located within the demarcated playing zone.
The game apparatus preferably also comprises an additional means for reversibly holding at least one positive unit and an additional means for reversibly holding at least one negative unit. More preferably, the game apparatus further comprises an additional means for reversibly holding a plurality of positive units and an additional means for reversibly holding a plurality of negative units. In one embodiment of the invention, the additional holding means comprises a groove divided into two sections, with one section adopted to hold a plurality of the positive units and the other section adopted to hold a plurality of the negative units. Most preferably, the additional positive unit holding means and the additional negative unit holding means are an integral part of the playing surface and are located outside the demarcated playing zone.
The playing surface of the game apparatus can be a screen of an electrical unit (such as a computer screen, a television screen, etc), a surface of a game board, a surface of a blackboard or other writing surface, a Velcro(copyright) surface, a static electricity charged surface, a magnetized surface, a magnetizable surface, or the surface of any other apparatus or device capable of displaying the demarcated playing zone, the null units, the positive units, the negative units, and the means for showing the number of free positive units and/or free negative units within the demarcated playing zone.
The game apparatus of the present invention is employed in conjunction with methods that utilize the principles of the Null Theory. In particular, in the case of adding and subtracting positive and negative numbers, the method of the present invention comprises the following steps:
Step A:
Play begins in a null state where a demarcated playing zone comprises a plurality of null units, with each null unit comprising at least one positive unit reversibly attached to at least one negative unit, the number of positive and negative units per null unit being equal.
Step B:
Take the first mathematical operation in a mathematical problem X1 Sm Xm Sn Xn . . . Sz Xz where the mathematical expressions X1, Xm, Xn, and Xare independently selected from the group consisting of positive numbers, negative numbers, and combinations thereof, the mathematical operators Sm, Sn, and Sz are independently selected from the group consisting of the addition operation and the subtraction operation, m is selected from the group consisting of 0 and 2, n is selected from the group consisting of 0 and 3, provided that if m is 0, n is 0, and z is selected from the group consisting of 0 and whole integers greater that 3, provided that if m is 0, z is 0, and perform the first mathematical operation X1 as follows:
1. If X1 is a positive number, then the mathematical operation of step (B) comprises moving or adding X1 free positive units to the demarcated playing zone.
2. If X1 is a negative number, then the mathematical operation of step (B) comprises moving or adding the absolute value of X1 free negative units to the demarcated playing zone.
Alternatively, the mathematical operation indicated by the first mathematical expression X1 can be performed as follows:
1. If X1 is a positive number, then the mathematical operation of step (B) can be accomplished by breaking apart or separating X1, null units that are within the demarcated playing zone into X1 free positive units and into X1, free negative units and removing or subtracting the X1 free negative units from the demarcated playing zone.
2. If X1 is a negative number, then the mathematical operation of step (B) can be accomplished by breaking apart or separating the absolute value of X1 null units that are within the demarcated playing zone into the absolute value of X1 free positive units and into the absolute value of X1 free negative units and removing or subtracting the absolute value of X1 free positive units from the demarcated playing zone.
Step C:
Take the second mathematical operation S2X2 in the mathematical problem, where S2 is selected from the group consisting of the addition operation and the subtraction operation and X2 is selected from the group consisting of positive and negative numbers, and perform the mathematical operation indicated by S2X2 as follows:
1. When S2 is an addition operation and X2 is a positive number (e.g., +(+3)), then step (C) comprises moving or adding X2 free positive units to the demarcated playing zone and, if there are any free negative units within the demarcated playing zone, then step (C) further comprises combining up to X2 free negative units that are already inside the demarcated playing zone with up to the X2 free positive units that were moved into the demarcated playing zone.
2a. When S2 is a subtraction operation and X2 is a positive number (e.g., xe2x88x92(+3)), then step (C) comprises removing or subtracting X2 free positive units from the demarcated playing zone and, if there are not X2 free positive units within the demarcated play zone to remove from the demarcated zone, then step (C) further comprises breaking apart or separating enough null units that are within the demarcated playing zone to obtain up to the required X2 free positive units and removing the X2 free positive units from the demarcated playing zone.
2b. Alternatively, when S2 is a subtraction operation and X2 is a positive number (e.g., xe2x88x92(+3)), then step (C) can also be accomplished by moving or adding X2 free negative units to the demarcated playing zone and, if there are any free positive units within the demarcated playing zone, then step (C) further comprises combining up to X2 free positive units that are already inside the demarcated playing zone with up to the X2 free negative units that were moved into the demarcated playing zone.
3a. When S2 is an addition operation and X2 is a negative number (e.g., +(xe2x88x923)), then step (C) comprises moving or adding the absolute value of X2 free negative units to the demarcated playing zone and, if there are any free positive units within the demarcated playing zone, then step (C) further comprises combining up to the absolute value of X2 free positive units that are already within the demarcated playing zone with up to the absolute value of X2 free negative units that were moved into the demarcated playing zone.
3b. Alternatively, when S2 is an addition operation and X2 is a negative number (e.g., +(xe2x88x923)), then step (C) can also be accomplished by removing or subtracting X2 free positive units from the demarcated playing zone and, if there are not X2 free positive units within the demarcated play zone to remove from the demarcated zone, then step (C) further comprises breaking apart or separating enough null units that are within the demarcated playing zone to obtain up to the required X2 free positive units and removing the X2 free positive units from the demarcated playing zone.
4. When S2 is a subtraction operation and X2 is a negative number (e.g., xe2x88x92(xe2x88x923)), then step (C) comprises removing or subtracting the absolute value of X2 free negative units from the demarcated playing zone and, if there are not enough absolute value of X2 free negative units within the demarcated play zone to remove from the demarcated playing zone, then step (C) further comprises breaking apart or separating enough null units that are within the demarcated playing zone to obtain up to the required absolute value of X2 free negative units and removing the absolute value of X2 free negative units from the demarcated playing zone.
Step D:
Repeat step (C) for each of the remaining mathematical operations SnXn through SzXz in the problem, where Xn through Xz are independently selected from the group consisting of positive numbers, negative numbers, and combinations thereof, and the mathematical operators Sn through Sz are independently selected from the group consisting of the addition operation and the subtraction operation.
As the above discussion of adding and subtracting positive and negative numbers demonstrates, when the operation sign and the sign of the mathematical expression are different, the same result follows whether the operation sign or the sign of the mathematical expression is treated as the mathematical operator as long as the other sign is treated as the sign of the unit. To illustrate, for the mathematical expression xe2x88x92(+3), the same result is obtained when the xe2x80x9cxe2x88x92xe2x80x9d sign is treated as the mathematical operator and the xe2x80x9c+xe2x80x9d sign is treated as designating the type of unit (i.e., 3 positive units) as when the xe2x80x9c+xe2x80x9d sign is treated as the mathematical operator and the xe2x80x9cxe2x88x92xe2x80x9d sign is treated as designating the type of unit (i.e., 3 negative units). This principal of interchangeability of signs, which is also applicable to multiplying and dividing positive and negative numbers, follows from the well established rule that ab=ba or, in our specific case, (+)(xe2x88x92)=(xe2x88x92)(+). Accordingly, students learning how to add, subtract, multiply, and divide positive and negative numbers do not have to memorize which sign is to be treated as the operator sign and which sign has to be treated as the unit sign. The students just have to remember that if they treat one of the signs as the operator sign, they must treat the other sign as the unit sign.
In the case of multiplying positive and negative numbers, the method of the present invention comprises the following steps:
Step A:
Play begins in a null state where a demarcated playing zone comprises a plurality of null units, with each null unit comprising at least one positive unit reversibly attached to at least one negative unit, the number of positive and negative units per null unit being equal.
Step B
In this scenario, X1 is a multiplication expression (SxM1)(SyN1), where Sx and Sy are independently selected from the group consisting of a positive sign and a negative sign, M1 is the absolute value of a number, and N1 is the absolute value of a number. The multiplication expression (SxM1)(SyN1) is performed as follows:
1. If Sx and Sy are positive signs, move or add the absolute value of (M1)(N1) positive units to the demarcated playing zone. Since the demarcated playing zone is initially in the null state, there are initially no free positive units or free negative units in the demarcated playing zone. Accordingly, after adding the absolute value of (M1)(N1) positive units to the demarcated playing zone as required by step (B)(1), the total number of free positive units in the demarcated playing zone is the absolute value of (M1)(N1).
2a. If Sx is a positive sign and Sy is a negative sign, remove or subtract the absolute value of (M1)(N1) positive units from the demarcated playing zone. Since the demarcated playing zone is initially in the null state, the free positive units are removed from the demarcated playing zone by taking the absolute value of (M1)(M1) null units that are within the demarcated playing zone, breaking or separating them into their constituent positive units and negative units, and removing the absolute value of (M1)(N1) positive units from the demarcated playing zone. Thus, the absolute value of (M1)(N1) negative units are left behind in the demarcated playing zone.
2b. Alternatively, when Sx is a positive sign and Sy is a negative sign, then step (B) can also be accomplished by moving or adding the absolute value of (M1)(N1) negative units to the demarcated playing zone. Since the demarcated playing zone is initially in the null state, there are initially no free positive units or free negative units in the demarcated playing zone. Accordingly, after adding the absolute value of (M1)(N1) negative units to the demarcated playing zone as required by step (B)(2b), the total number of free negative units in the demarcated playing zone is the absolute value of (M1)(M1).
3a. If Sx is a negative sign and Sy is a positive sign, move or add the absolute value of (M1)(N1) negative units to the demarcated playing zone. Since the demarcated playing zone is initially in the null state, there are initially no free positive units or free negative units in the demarcated playing zone. Accordingly, after adding the absolute value of (M1)(M1) negative units to the demarcated playing zone as required by step (B)(3a), the total number of free negative units in the demarcated playing zone is the absolute value of (M1)(M1).
3b. Alternatively, when Sx is a negative sign and Sy is a positive sign, then step (B) can also be accomplished by removing or subtracting the absolute value of (M1)(M1) positive units from the demarcated playing zone. Since the demarcated playing zone is initially in the null state, the free positive units are removed from the demarcated playing zone by taking the absolute value of (M1)(N1) null units that are within the demarcated playing zone, breaking or separating them into their constituent positive units and negative units, and removing the absolute value of (M1)(M1) positive units from the demarcated playing zone. Thus, the absolute value of (M1)(N1) negative units are left behind in the demarcated playing zone.
4. If Sx and Sy are negative signs, remove or subtract the absolute value of (M1)(N1) negative units from the demarcated playing zone. Since the demarcated playing zone is initially in the null state, the free negative units are removed from the demarcated playing zone by taking the absolute value of (M1)(N1) null units that are within the demarcated playing zone, breaking or separating them into their constituent positive units and negative units, and removing the absolute value of (M1)(N1) negative units from the demarcated playing zone. Thus, the absolute value of (M1)(N1) positive units are left behind in the demarcated playing zone.
In the case of dividing positive and negative numbers, the method of the present invention comprises the following steps:
Step A:
Play begins in a null state where a demarcated playing zone comprises a plurality of null units, with each null unit comprising at least one positive unit reversibly attached to at least one negative unit, the number of positive and negative units per null unit being equal.
Step B
In this scenario, X1 is a division expression (SxxM1)/(SyyN1), where Sxx and Syy are independently selected from the group consisting of a positive sign and a negative sign, M1 is the absolute value of a number, and N1 is the absolute value of a number. The division expression (SxxM1)/(SyyN1) is performed as follows:
1. If Sxx and Syy are positive signs, move or add the absolute value of (M1)/(N1) positive units to the demarcated playing zone. Since the demarcated playing zone is initially in the null state, there are initially no free positive units or free negative units in the demarcated playing zone. Accordingly, after adding the absolute value of (M1)/(N1) positive units to the demarcated playing zone as required by step (B)(1), the total number of free positive units in the demarcated playing zone is the absolute value of (M1)/(M1).
2a. If Sxx is a positive sign and Syy is a negative sign, remove or subtract the absolute value of (M1)/(N1) positive units from the demarcated playing zone. Since the demarcated playing zone is initially in the null state, the free positive units are removed from the demarcated playing zone by taking the absolute value of (M1)/(N1) null units that are within the demarcated playing zone, breaking or separating them into their constituent positive units and negative units, and removing the absolute value of (M1)/(M1) positive units from the demarcated playing zone. Thus, the absolute value of (M1)/(M1) negative units are left behind in the demarcated playing zone.
2b. Alternatively, when Sxx is a positive sign and Syy is a negative sign, then step (B) can also be accomplished by moving or adding the absolute value of (M1)/(M1) negative units to the demarcated playing zone. Since the demarcated playing zone is initially in the null state, there are initially no free positive units or free negative units in the demarcated playing zone. Accordingly, after adding the absolute value of (M1)/(M1) negative units to the demarcated playing zone as required by step (B)(2b), the total number of free negative units in the demarcated playing zone is the absolute value of (M1)/(M1).
3a. If Sxx is a negative sign and Syy is a positive sign, move or add the absolute value of (M1)/(N1) negative units to the demarcated playing zone. Since the demarcated playing zone is initially in the null state, there are initially no free positive units or free negative units in the demarcated playing zone. Accordingly, after adding the absolute value of (M1)/(N1) negative units to the demarcated playing zone as required by step (B)(3), the total number of free negative units in the demarcated playing zone is the absolute value of (M1)/(N1).
3b. Alternatively, when Sxx is a negative sign and Syy is a positive sign, then step (B) can also be accomplished by removing or subtracting the absolute value of (M1)/(N1) positive units from the demarcated playing zone. Since the demarcated playing zone is initially in the null state, the free positive units are removed from the demarcated playing zone by taking the absolute value of (M1)/(N1) null units that are within the demarcated playing zone, breaking or separating them into their constituent positive units and negative units, and removing the absolute value of (M1)/(N1) positive units from the demarcated playing zone. Thus, the absolute value of (M1)/(N1) negative units are left behind in the demarcated playing zone. ps 4. If Sxx and Syy are negative signs, remove or subtract the absolute value of (M1)/(N1) negative units from the demarcated playing zone. Since the demarcated playing zone is initially in the null state, the free negative units are removed from the demarcated playing zone by taking the absolute value of (M1)/(N1) null units that are within the demarcated playing zone, breaking or separating them into their constituent positive units and negative units, and removing the absolute value of (M1)/(N1) negative units from the demarcated playing zone. Thus, the absolute value of (M1)/(N1) positive units are left behind in the demarcated playing zone.
When the mathematical equation includes the addition and/or subtraction of one or more multiplication and/or division expressions by themselves and/or together with the addition and/or subtraction of one or more positive and/or negative numbers, each operation can be solved in the order that it appears in the equation. Alternatively, each of the multiplication and division expressions in the mathematical equation can be solved first, thereby converting the mathematical equation into one containing only addition and/or subtraction operations, and solving the resulting mathematical equation as discussed above.
As noted with respect to the game apparatus, in the methods of the present invention, each null unit preferably comprises one positive unit reversibly attached to or associated with one negative unit.
It is also preferred that the methods further comprise the step of measuring the number of the free positive units and the number of the free negative units within the demarcated playing zone. In one preferred version of the methods of the present invention, the measurement is performed by the step of placing the free positive units that are within the demarcated playing zone along the positive portion of an axis marked with substantially equal spaces from 0 to M and the step of placing the free negative units that are within the demarcated playing zone along the negative portion of an axis marked with substantially equal spaces from 0 to N, where M and N are as defined above. While the positive portion of the axis preferably forms a continuum with the negative portion of the axis, the present invention includes the embodiment where there is a separate positive axis having substantially equal spaces from 0 to M and a separate negative axis having substantially equal spaces from 0 to N, where M and N are as previously defined.
Furthermore, it is preferred that the methods of the present invention also comprise the step of measuring the number of free positive units and the number of free negative units that are to be added to or that have been removed from the demarcated playing zone. In one desirable version of the methods of the present invention, the measurement is performed by the step of placing the free positive units or the free negative units that are to be added to or that have been taken out of the demarcated playing zone along a second axis that is located outside the demarcated playing zone. The second axis is marked with substantially equal spaces from 0 to P, where P is as previously defined.
While the foregoing apparatuses and methods can be used to teach young children who are just learning to add and subtract only positive numbers, when teaching the addition and subtraction of only positive numbers, it is sufficient to use a simpler game apparatus such as one comprising (a) at least one means for measuring unit increments, (b) a plurality of means for indicating a single unit, and (c) a means for holding the plurality of single unit indicating means in slideable relationship to the measuring means. For instance, the plurality of single unit indicating means can comprise a plurality of beads, the holding means can comprise a dowel, with the beads being slideably mounted on the dowel, and the game apparatus can comprises two measuring means (such as two rulers), with each measuring means being position so that the game apparatus can be played with equal facility by both right and left handed players.